Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 112, 16 pages      arXiv:1107.5916      http://dx.doi.org/10.3842/SIGMA.2011.112
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs

Andrey V. Sokolov
V.A. Fock Department of Theoretical Physics, Sankt-Petersburg State University, 198504 St. Petersburg, Russia

Received August 06, 2011, in final form November 25, 2011; Published online December 05, 2011

Abstract
This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.

Key words: non-Hermitian quantum mechanics; supersymmetry; exceptional points; resolution of identity.

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References

  1. Andrianov A.A., Sokolov A.V., Resolutions of identity for some non-Hermitian Hamiltonians. I. Exceptional point in continuous spectrum, SIGMA 7 (2011), 111, 19 pages, arXiv:1107.5911.
  2. Gel'fand I.M., Vilenkin N.J., Generalized functions, Vol. 4, Some applications of harmonic analysis, Academic Press, New York, 1964.
  3. Sokolov A.V., Andrianov A.A., Cannata F., Non-Hermitian quantum mechanics of non-diagonalizable Hamiltonians: puzzles with self-orthogonal states, J. Phys. A: Math. Gen. 39 (2006), 10207-10227, quant-ph/0602207.
  4. Andrianov A.A., Cannata F., Sokolov A.V., Spectral singularities for non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity, J. Math. Phys. 51 (2010), 052104, 22 pages, arXiv:1002.0742.

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